Daniel Lichtblau <danl at wolfram.com> wrote in message news:<c6vhrn$gcq$1 at smc.vnet.net>...
> AC wrote:
The core of the disagreement seems to be that you interpret decimal
numbers as approximate and I don't. I challenge you to provide a
mathematical proof that 1.3 differs from 13/10 without any reference
to computers. After all the decimal arithmetic was invented and widely
used long before computers.
>
> By the way: What is 3 + 1.65? What is 1.3 + 1.3*10^(-100)? What is Pi +
> 1.65? What is 1.3*Pi? How do you propose these be represented?
>
3 + 1.65 === 4.65===93/20
1.3 + 1.3*10^(-100)===
130000000000000000000000000000000000000000000000000000000000000000000000000000\
000000000000000000000013/\
100000000000000000000000000000000000000000000000000000000000000000000000000000\
000000000000000000000000
Pi + 1.65 === Pi + 165/100 === Pi + 33/20
1.3*Pi === 13*Pi/10
I completely agree that
N[1.3*Pi] => 4.084070449666731
or
N[1.3 + 1.3*10^(-100)]=> 1.300000000000000
> Any serious follow up will address these examples in detail.
>
>
> > [...]
> > You apparently have problems to distiguish number from its internal
> > Mathematica representation. It seem like you don't care what a user
> > may need.
>
> I rather doubt I have any such problem, but that's not the issue.
This is precisely the issue.
> As has been noted in this thread, Mathematica works with approximate
> numbers in binary representation, and if the input is approximate
Mathematica also works with exact expressions and treating 1.3 as an
approximate number is completely ARBITRARY decision made by its
developers. What is wrong with a variation of the notation to
explicitly define approximate numbers such as ~1.3 or 1.3...?
> then that's the end of the story. This is true regardless of how
> you might choose to think of the numbers: exact, approximate, or
> living is some way-cool realm of Heisenberg uncertainty in regard
> to those possibilities.
The results of physical experiments or some numerical computations are
presented as decimal numbers and often approximate indeed. But that is
not a reason to claim that some decimals are always approximate and
some not. I am referring here to the fact that some numbers such as
1.25 do have finite binary representation. Think about it, in your
interpretation 0.25 is in essence an exact number but 0.35 not. That's
mathematical nonsense.
The Heisenberg principle has nothing to do with the discussed topic;
you just unnecessarily exposed your lack of knowledge.
AC
PS
I refer you to a new thread just started '34.123*89 = 3036.95
(3036.947)' here on the mathgroup, so much about your approach being
intuitive and in the mainstream.
>
[... much of repetive arguments deleted ..]
>
> Daniel Lichtblau
> Wolfram Research